Linear equations are algebraic expressions that describe the relationship between variables linearly. These equations are considered the simplest form of equations involving addition, subtraction, and multiplication. The standard form is:

\[a_1x_1 + a_2x_2 + \ldots + a_nx_n = b\]

  • \(x_1, x_2, \ldots, x_n \in \mathbb{R} \) are the variables.
  • \(a_1, a_2, \ldots, a_n \in \mathbb{R} \) and \(a \neq 0\), are the coefficients.
  • \(b \in \mathbb{R} \) is the constant term.

A linear equation of the first degree can be expressed in the form \(ax = b\), which has a unique solution, given by:

\[x= \frac{b}{a}\]

It is important to note that since \(a \neq 0\), by definition, a linear equation always admits one and only one solution.

Solving first-degree equations can be straightforward. If the initial equation is already in its standard form, the process typically involves a series of steps to isolate the variable and perform the necessary calculations.

Example

Solve the equation \(2x + 3 = 11x\)

As the first step, we brought the equation into its standard form. In this case, there are no restrictions on the value of \(x\), which exists throughout the domain of real numbers \(x \in \mathbb{R}\). By performing the calculations, we obtain:

\begin{align*} 2x + 3 &= 11x \\[0.6em] 2x – 11x &= -3 \\[0.6em] -9x &= -3 \\[0.6em] x &= \frac{3}{9} \\[0.6em] x &= \frac{1}{3} \end{align*}


Substituting the value \(x = \large{\frac{1}{3} }\) into the original equation is crucial to ensuring the solution’s accuracy. Based on the calculations performed, the final equality is indeed confirmed.

\begin{align*} 2\left(\frac{1}{3}\right) + 3 &= 11\left(\frac{1}{3}\right)\\[0.6em] \frac{2}{3} + 3 &= \frac{11}{3} \\[0.6em] \frac{2}{3} + \frac{9}{3} &= \frac{11}{3} \\[0.6em] \frac{11}{3} &= \frac{11}{3} \\[0.6em] \end{align*}

The solution to the equation is:

\[ x = \frac{1}{3} \]

It is important to remember that a first-degree linear equation is a polynomial equation, which means it must be expressed as the sum or difference of monomials. For instance, an equation that involves a variable \(x\) in the denominator is not linear and is, therefore, non-linear. The following equation is not linear:

\[ \frac{1}{x}+3 = 0\]

It is considered a rational equation because it involves a fraction with the variable \(x\) in the denominator. To solve this equation, we must restrict the permissible values of the variable \(x\). In the case of this example, \(x\) can take all values that belong to the field of real numbers except for zero. In formal terms, the equation is defined \(\forall x \in \mathbb{R} \neq {0}\)


Geometrical meaning

Linear equations in two variables represent geometrically a lines on the Cartesian plane and can be expressed in the form of:

\[y = mx + b\]

  • \(m\) is a coefficient that represents the slope of the line.
  • \(b\) is a coefficient that represents the point of intersection on the y-axis, known as the y-intercept.

The solutions to these equations provide the coordinates of the points where the lines intersect the axes. For example, the line with equation \(y = x + 3\) is represented on the Cartesian plane as follows:

We have: \(y = x + 3\).
The line intersects the \(y\)-axis at \(3\).

We have: \(y = 3\).
The line is parallel to the x-axis at \(y = 3\).

Expressions of the form \(x = 3\) or \(y = 3\) represent lines with a constant value, parallel respectively to the y-axis and the x-axis.